In this geometry problem, three circles are mutually tangent to each other and also tangent to a common line \( L \). The radii of circles \( A \) and \( B \) are given, and we determine the radius of circle \( C \).
Problem Statement
Three circles are tangent to each other and to a line \( L \). The radius of circle \( A \) is \( 10 \text{ cm} \), and the radius of circle \( B \) is \( 8 \text{ cm} \). Find the radius of circle \( C \).
Step-by-Step Solution
Let the radii of the three circles be:
\[ a = 10, \quad b = 8, \quad c = ? \]Draw perpendiculars from the centers of the circles to line \( L \). Applying the Pythagorean theorem to the appropriate right triangles allows us to determine horizontal distances between the centers.
Step 1: Triangle \( BCB' \)
Using the Pythagorean theorem:
\[ x^2 + (b - c)^2 = (b + c)^2 \]Expanding:
\[ x^2 + b^2 - 2bc + c^2 = b^2 + 2bc + c^2 \]Simplifying:
\[ x^2 = 4bc \implies x = 2\sqrt{bc} \]Step 2: Triangle \( ACA' \)
Expanding and simplifying:
\[ y^2 = 4ac \implies y = 2\sqrt{ac} \]Step 3: Triangle \( ABA'' \)
Simplifying:
\[ z^2 = 4ab \implies z = 2\sqrt{ab} \]Step 4: Using the Geometric Relationship
Since the horizontal distance satisfies:
\[ z = x + y \]Substitute the expressions:
\[ 2\sqrt{ab} = 2\sqrt{bc} + 2\sqrt{ac} \]Divide both sides by 2 and factor \( \sqrt{c} \):
\[ \sqrt{ab} = \sqrt{bc} + \sqrt{ac} = \sqrt{c}(\sqrt{b} + \sqrt{a}) \]Solve for \( c \):
\[ c = \frac{ab}{(\sqrt{a} + \sqrt{b})^2} \]Step 5: Substitute the Given Values
Final Answer
\[ \boxed{c \approx 2.2 \text{ cm}} \]Therefore, the radius of circle \( C \) is approximately 2.2 cm.